Generalized Hilbert Operator Acting on Weighted Bergman Spaces and on Dirichlet Spaces
Let $μ$ be a positive Borel measure on the interval [0,1). For $β> 0$, The generalized Hankel matrix $\mathcal{H}_{μ,β}= (μ_{n,k,β})_{n,k\geq0}$ with entries $μ_{n,k,β}= \int_{[0.1)}\frac{Γ(n+β)}{n!Γ(β)} t^{n+k}dμ(t)$, induces formally the operator $$\mathcal{H}_{μ,β}(f)(z)=\sum_{n=0}^\infty \left(\sum_{k=0}^\infty μ_{n,k,β}a_k\right)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. In this paper, we characterize those positive Borel measures on $[0,1)$ such that $\mathcal{H}_{μ,β}(f)(z)= \int_{[0,1)} \frac{f(t)}{(1-tz)^β} dμ(t)$ for all in weighted Bergman Spaces $A_α^p(0 -1)$, and among them we describe those for which $\mathcal{H}_{μ,β}(β>0)$ is a bounded(resp.,compact) operator on weighted Bergman spaces and Dirichlet spaces.